Statistica Sinica 26 (2016), 157-175

doi:http://dx.doi.org/10.5705/ss.2014.064

WEIGHTED ANGLE RADON TRANSFORM:

CONVERGENCE RATES AND EFFICIENT ESTIMATION

Daniel Hohmann and Hajo Holzmann

Philipps-University Marburg

Abstract: In the statistics literature, recovering a signal observed under
the Radon transform is considered a mildly ill-posed inverse problem. In
this paper, we argue that several statistical models that involve the Radon
transform lead to an observational design which strongly influences its
degree of ill-posedness, and that the Radon transform can actually become
severely ill-posed. The main ingredient here is a weight function λ on the
angle. Extending results for the limited angle situation, we compute the
singular value decomposition of the Radon transform as an operator between
suitably weighted L_{2}-spaces, and show how the singular values relate to
the eigenvalues of the sequence of Toeplitz matrices of λ. Further, in the
associated white noise sequence model, we give upper and lower bounds on
the rate of convergence, and in several special situations even obtain optimal
rates with precise minimax constants. For the severely ill-posed limited angle
problem, a simple projection estimator is adaptive in the exact minimax
sense.

Key words and phrases: Efficient estimation, limited angle problem, minimax estimation, nonparametric estimation, Radon transform.